Graphs on Surfaces with Positive Forman Curvature Or Corner Curvature

GRAPHS ON SURFACES WITH POSITIVE FORMAN CURVATURE OR CORNER CURVATURE

YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG

Abstract. On one hand, we study the class of graphs on surfaces, satisfying tessellation properties, with positive Forman curvature on each edge. Via medial graphs, we provide a new proof for the ﬁniteness of the class, and give a complete classiﬁcation. On the other hand, we classify the class of graphs on surfaces with positive corner curvature.

Contents 1. Introduction 1 2. Preliminaries 5 2.1. Embedding 9 3. Graphs on surfaces with positive Forman curvature 10 4. Graphs on surfaces with positive corner curvature 13 References 20

1. Introduction The Gaussian curvature of a smooth surface is well studied in differential geometry, which describes the convexity of the surface. For a polyhedron in R3, the discrete Gaussian curvature, as a measure, concentrates on the set of vertices. In the graph theory, the combinatorial curvature of a planar graph, which serves as the discrete Gaussian curvature of a canonical piecewise flat surface, was introduced by [31, 36, 17, 23] respectively. It has been extensively studied in the literature; see e.g. [38, 42, 20, 3, 18, 28, 21, 37, 34, 4, 13, 11, 40, 10, 25, 27, 26, 32, 16]. Let S be a (possibly noncompact) connected surface without boundary. Let (V,E) be a (possibly infinite) locally finite, undirected, simple graph with the set arXiv:2002.03550v1 [math.CO] 10 Feb 2020 of vertices V and the set of edges E. We may regard (V,E) as a 1-dimensional topological space. Let ϕ :(V,E) → S be an topological embedding. We denote by F the set of faces induced by the embedding ϕ, i.e. connected components of the complement of the embedding image of (V,E) in S. We write G = (V,E,F ) for the cellular complex structure induced by the embedding, which is called a graph on a surface [30] (or a semiplanar graph [22]). For any σ ∈ F, we denote by σ the closure of σ in S, which is called the closed face. We say that a semiplanar graph G = (V,E,F ) is a tessellation of S if the following hold, see e.g. [26]: (i) For any compact set K ⊂ S, K can be covered by finitely many closed faces. (ii) Every closed face is homeomorphic to a closed disk whose boundary consists of finitely many edges of the graph. 1 2 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG

(iii) Every edge is contained in exactly two different closed faces. (iv) If two closed faces intersect, then the intersection is either a vertex or the closure of an edge. One can show that a tessellation of S is finite if and only if S is compact. It is called a planar tessellation (resp. a tessellation in the real projective plane) if S is the sphere S2 or the plane R2 (resp. RP 2). In this paper, we always consider tessellations, and call planar tessellations planar graphs for simplicity. For a graph on a surface G = (V,E,F ), two vertices x, y are called neighbors if there is an edge connecting x and y, denoted by x ∼ y. Two elements in V,E,F are called incident if the closures of their embedding images intersect. If one is contained in the closure of the other, then we write like x ≺ e, e ≺ σ, x ≺ σ, for x ∈ V, e ∈ E, σ ∈ F. We denote by |x| (resp. |σ|) the degree of a vertex x (resp. a face σ), i.e. the number of neighbors of x (resp. the number of edges incident to σ). In this paper, we only consider semiplanar graphs satisfying the following: for any vertex x and face σ, |x| ≥ 3, |σ| ≥ 3. Given a graph G = (V,E,F ) on a surface S, the combinatorial curvature at a vertex x is defined as |x| X 1 Φ(x) = 1 − + . 2 |σ| σ∈F :x≺σ We endow S with a canonical piecewise flat metric as follows: assign each edge length one, replace each face by a regular Euclidean polygon of side-length one with same facial degree, and glue these polygons along the common edges; see [8] for gluing metrics. The ambient space S equipped with the gluing metric is called the (regular Euclidean) polyhedral surface of G, denoted by S(G). For the metric surface S(G), the generalized Gaussian curvature is a measure concentrated on vertices, whose mass at each vertex x is given by the angle defect K(x), i.e. 2π minus the total angle at x. One easily sees that 1 Φ(x) = K(x), ∀x ∈ V, 2π where K(x) is the angle defect, i.e. the discrete Gaussian curvature, at the vertex x. If S is compact, the discrete Gauss-Bonnet theorem reads as X (1) Φ(x) = χ(S), x∈V where χ(·) is the Euler characteristic of S. A planar graph G has nonnegative combinatorial curvature if and only if the polyhedral surface S(G) is a generalized convex surface in the sense of Alexandrov, see [9, 8, 22]. Higuchi [20] conjectured a discrete Bonnet-Myers theorem that a graph on a surface G with positive combinatorial curvature everywhere is finite. It was confirmed by DeVos and Mohar [13], see e.g. [11, 10, 32, 33, 16] for further developments. Theorem 1.1 ([13]). For a graph G = (V,E,F ) on a surface S, if Φ(x) > 0 for any x ∈ V, then G is finite. Moreover, S = S2 or RP 2. For a planar graph G with an embedding, one can define the dual graph G∗: the vertices of G∗ are corresponding to faces of G, the faces of G∗ are corresponding to vertices of G, and two vertices in G∗ are adjacent if and only if the corresponding GRAPHS ON SURFACES WITH POSITIVE FRRMAN OR CORNER CUVATURES 3 faces in G share a common edge. One can show that for a finite graph G, it is a tessellation if and only if so is G∗, see e.g. [4]. For a graph on a surface G with positive combinatorial curvature, the dual graph may not have nonnegative combinatorial curvature; 7-gonal prism (resp. 5-gonal antiprism (Figure 5 (right))) has positive combinatorial curvature everywhere, but the dual, the 7-gonal bipyramid (resp. 5- gonal pseudo-double wheel (Figure 8 (upper right))), has negative curvature at the two apexes. This means that the dual operation doesn’t preserve the class of graphs on surfaces with positive combinatorial curvature. We will study some curvature notions of graphs on surfaces, for which the dual operation is closed on the class of graphs with positive curvature. Motivated by Bochner techniques for differential forms in Riemannian geometry, Forman [15] introduced the curvature on general CW complexes, which is now called the Forman curvature. For a compact Riemannian manifold M, let ∆p be the Hodge Laplacian on p-forms, p ∈ N0. The Bochner-Weitzenböck formula reads as ∗ ∆p = (∇p) ∇p + Fp, where ∇p is the Levi-Civita covariant derivative operator on p-forms and Fp de- notes the curvature operator on p-forms. On a CW complex, Forman derived an analogous formula for the discrete Hodge Laplacian and defined the remainder term Fp as the discrete curvature on p-cells. In this paper, we only consider the Forman curvature F1 on 1-cells, i.e. edges, with weight one everywhere. This curvature F1 corresponds to the discrete analog of Ricci curvature, and we will denote it by RicF in this paper. For a graph on a surface G = (V,E,F ), two edges e1 and e2 are called parallel neighbors if one and only one of the following holds:

(1) There exists x ∈ V such that x ≺ e1, x ≺ e2. (2) There exists σ ∈ F such that e1 ≺ σ, e2 ≺ σ. The Forman curvature of an edge e is deﬁned as, see [15],

(2) RicF (e) = #{σ ∈ F : e ≺ σ}+#{x ∈ V : x ≺ e}−#{parallel neighbors of e}. We are interested in graphs on surfaces with positive Forman curvature (everywhere). Note that a graph on surface with positive Forman curvature may not be a tessellation, see e.g. Figure 1. There are inﬁnitely many nontessellation, pla-

2 Figure 1. The graph C4 embedded in S . Each edge e has RicF (e) = 3. nar graphs with positive Forman curvature (everywhere). For example, the star graph K1,n (n ≥ 1) has positive Forman curvature 3 everywhere. In this paper, we only study the class of tessellations on surfaces with positive Forman curvature, denoted by FC+. One easily checks that the class FC+ is closed under the dual operation. Forman proved discrete analogs of Bochner’s theorem and the Bonnet- Myers theorem for “quasi-convex” regular CW complexes, see [15, Theorem 2.8 and Theorem 6.3]. In our setting, for a graph G = (V,E,F ) on a surface S, they state 4 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG

1 that if RicF (e) > 0 for each edge e, then H (S, R) = 0 and the diameter of X is ﬁnite, respectively. The above theorems yield the following result. Theorem 1.2 ([15]). Let G = (V,E,F ) be a tessellation on a surface S satisfying 2 2 RicF (e) > 0 for all e ∈ E. Then G is ﬁnite and S = S or RP .

e x2 x1

Figure 2

One of main difficulties for the Bonnet-Myers theorem above is that the total Forman curvature of an infinite graph on a surface is possibly infinite. So that we don’t have the control for the number of edges in the case of positive Forman curvature directly. Forman circumvented the difficulty by using (combinatorial) Jacobi fields. In this paper, we give a new proof of the above result without using Jacobi fields. The proof strategy is as follows: for a tessellation G of a surface S, we obtain a formula of the Forman curvature of an edge e,

(3) RicF (e) = 16 − (|x1| + |x2| + |f1| + |f2|), where x1, x2 ∈ V and f1, f2 ∈ F such that x1 ≺ e, x2 ≺ e, e ≺ f1, e ≺ f2, see Figure 2 and Proposition 4. For G, we construct a medial graph G0, a graph on a surface S, associated to G, whose vertices correspond to the set of edges in G, identified with the midpoints of edges, and whose faces correspond to the set of vertices and faces in G, see Section 2 for details. For a graph G on a surface with positive Forman curvature, by the structure of medial graphs and (3), we will prove that the medial graph G0 has positive combinatorial curvature everywhere, see Proposition 2.6. By Theorem 1.1, G0 is a finite graph and the ambient space S is S2 or RP 2. This provides a new proof of the theorem. Via medial graphs, we classify the class FC+. By the discrete Gauss-Bonnet theorem, we estimate that the number of vertices in a medial graph of G in FC+ 2 2 S RP is at most 24, see Lemma 2.7. We denote by FC+ (resp. FC+ ) planar graphs 2 (resp. graphs on RP ) in FC+. Since the medial graph is 4-regular, we enumerate 2 S the medial graphs of planar graphs in FC+ using the algorithm of spherical quad- 2 rangulations by Brinkmann et al. [6]. For the graphs on RP in FC+, we use the 2 S 2 2 classification of FC+ and the properties of the double covering map S → RP .

Theorem 1.3. There are 116 graphs in FC+ up to isomorphism, which are all 2 RP planar graphs, i.e. FC+ = ∅. In the last part of the paper, we consider the corner curvature for graphs on surfaces. Baues and Peyerimhoﬀ [3] introduced the so-called corner curvature for graphs on surfaces. A corner of G = (V,E,F ) is a pair (x, σ) such that x ∈ V, σ ∈ F and x is incident to σ. The corner curvature of a corner (x, σ) is deﬁned as 1 1 1 C(x, σ) := + − . |x| |σ| 2 GRAPHS ON SURFACES WITH POSITIVE FRRMAN OR CORNER CUVATURES 5

Baues and Peyerimhoﬀ [3, 4] proved many interesting properties, e.g. a discrete Cartan-Hadamard Theorem, for planar graphs with non-positive corner curvature. We denote by C+ := {G : C(x, σ) > 0, ∀ corner (x, σ)} the class of graphs on surfaces with positive corner curvature. It turns out the corner curvature condition is quite strong in general, e.g. for any G ∈ C+, it P has positive combinatorial curvature, by Φ(x) = x≺σ C(x, σ). Since the dual operation switches the vertices and faces of a graph on a surface, one can show that ∗ G ∈ C+ if and only if G ∈ C+; see [4]. By the discrete Gauss-Bonnet theorem and some combinatorial arguments, we prove that the number of vertices of a planar graph with positive corner curvature, or of its dual graph, is at most 12. Then we modify Brinkmann-McKay’s program plantri [7] to classify the set of planar graphs with positive corner curvature. It is well-known for the community that C+ is not a large class. In fact, Keller classiﬁed the class C+ using hand calculation, according to private communication to him. Theorem 1.4. There are 22 planar graphs and 2 graphs in the projective plane with positive corner curvature up to isomorphism. The programs are available via https://github.com/akmyh2/PFCPCnC/. The rest of the paper is organized as follows: In the next section, we review embedding of graphs to the plane. In Section 3 and Section 4, we explain how to enumerate all the FC+- and C+-graphs.

Acknowledgements. We cordially thank Gunnar Brinkmann, Beifang Chen, Nico van Cleemput, Matthias Keller, Shiping Liu, Brendan McKay, Min Yan, Flo- rentin Münch, and Norbert Peyerimhoff for many helpful discussions on curvatures on planar graphs. A. is supported by JSPS KAKENHI Grant Number JP16K05247. The work was done when the first author was visiting School of Mathematical Sciences, and Shanghai Center for Mathematical Sciences, Fudan University. He thanks for the hospitality of these institutes. H. is supported by NSFC, no.11831004 and no. 11926313. S. is supported by NSFC grant no. 11771083 and NSF of Fuzhou University through grant GXRC-18035.

2. Preliminaries Let G = (V,E,F ) be a graph on a surface S. Usually, we do not distinguish V,E,F with their embedding image in S. We recall basic results in graph theory. Proposition 2.1 ([4]). For a ﬁnite graph G, it is a tessellation if and only if so is G∗. A graph (embedded or not) is said to be k-connected if it cannot be disconnected by removing fewer than k vertices. Proposition 2.2. Any ﬁnite planar tessellation G = (V,E,F ) is 2-connected. Proof. For any v ∈ V, we want to show that G is connected if we remove the N vertex v and its incident edges. Let {σi}i=1 be the set of faces incident to v. Then N by the tessellation properties, ∪i=1σi is homeomorphic to a closed disk, and its boundary γ consists of edges, which is a simple closed curve. By the Jordan curve 2 theorem, S \ γ consists of two disjoint open disks D1 and D2. Without loss of 6 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG

N generality, we assume that ∪i=1σi ⊂ D1. Then D2 ∪ γ is connected, which implies the connectedness of the graph G by removing v and its incident edges. For our purposes, we need the notion of the medial graph of G, introduced by Steinitz [35], see e.g. [5, p. 104]. The medial graph G0 = (V 0,E0,F 0) of G = (V,E,F ) is deﬁned as follows: For each e ∈ E, choose an interior point (say mid- 0 point) Me on the edge e. Let V be the set of Me for all e ∈ E. The vertices Me1 and

Me2 are adjacent if there exist x ∈ V, f ∈ F such that x ≺ e1, x ≺ e2, e1 ≺ f, e2 ≺ f.

For any f ∈ F, draw a curve ce1,e2 on f between any adjacent Me1 and Me2 on the 0 boundary of f, such that ce1,e2 does not cross each other. Let E be the set of such curves, F 0 be the set of connected components of S \(V 0 ∪E0). As in Figure 4, each f 0 ∈ F 0 corresponds to either f ∈ F or x ∈ V , and G0 is 4-regular. For example, the medial graph of tetrahedron is the graph of the octahedron. The medial graph of octahedron or cube is the graph of the cuboctahedron (Figure 3), where the cuboctahedron is an Archimedean solid. More generally, the medial graph of the graph of a Platonic solid P is the graph of a daughter polyhedron (Conway, Burgiel, and Goodman-Strauss [12, Sect. 21 “Naming Archimedean and Catalan Polyhedra and Tilings”, p. 285]) of P and the dual Q. Medial graphs appear tacitly in [38]. A tessellation G = (V,E,F ) satisfies a strong isoperimetric inequality, if the average of combinatorial curvature X 1 X 1 Ψ(e) = + − 1 (e ∈ E) |x| |f| x ≺ e e ≺ f x ∈ V f ∈ F of medial graph A0 of finite induced subgraph A of G has negative supremum as A grows toward G. The same argument as in Proposition 2.1 in [1] yields the following result. Proposition 2.3 ([1]). If G is a tessellation, then so is the medial graph G0. It is obvious that a finite graph G on surface and its dual graph has same medial graphs, i.e. G0 = (G∗)0. Given a 4-regular graph on a surface, we want to figure out whether it is a medial graph of some graph on a surface. The following theorem is very useful, and there is a canonical way to construct the “inverse” medial graph. Theorem 2.4 ([2, Theorem 2.1]). Any embedded 4-regular graph whose faces can be 2-colored is the medial graph of a unique dual pair of embedding graphs.

For a 4-regular tessellation H = (V 0,E0,F 0) of S2, the unique dual pair (G, G∗) 0 0 0 0 0 0 of H is computed as follows: For f1, f2 ∈ F , we write f1 ∼ f2, if f1 is right 0 0 0 across from f2 with respect to some 4-valent v ∈ V . Then the equivalence relation ≈ ⊆ F 0 ×F 0 generated from ∼ has index two. For example, when H is the graph of cuboctahedron (Figure 3), the equivalence classes of ≈ are the set A1 of eight black 0 triangular faces and the set A2 of six white square faces. Let {A1,A2} be F / ≈. We associate Gi = (V,E,F ) as follows: Choose an equivalence class Ai, and an 0 0 inner point Pf 0 for each f ∈ Ai. Let V be the set of Pf 0 ’s. If a face f1 of Ai is right 0 0 0 0 0 across from a face f2 of Ai with respect to a 4-regular vertex v ∈ f1 ∩ f2 of V , 0 0 i.e. f ∼ f , then we draw exactly one simple curve a 0 0 from P 0 to P 0 through 1 2 f1,f2 f1 f2 v0. Let E be the set of such curves. Let F be the set of connected components of S2 \ (V ∪ E). GRAPHS ON SURFACES WITH POSITIVE FRRMAN OR CORNER CUVATURES 7

Figure 3. The cuboctahedron. The graph G0 = (V 0,E0,F 0) of cuboctahedron is the medial graph of the graph of cube G = (V,E,F ). V 0 corresponds to E. The 6 white square faces of G0 correspond to F , and the other 8 triangular faces of G0 to V .

If H is cuboctahedron, G1 (resp. G2) is a cube (resp. regular octahedron) when V = A1 (resp. A2). For a tessellation, we give a new formula for the Forman curvature. Proposition 2.5. Let G = (V,E,F ) be a tessellation on a surface S. For any e ∈ E, let x1, x2 ∈ V and f1, f2 ∈ F satisfy x1 ≺ e, x2 ≺ e, e ≺ f1, e ≺ f2. Then

(4) RicF (e) = 16 − (|x1| + |x2| + |f1| + |f2|).

Proof. By the tessellation properties, the closed faces f1 and f2 intersect only at the closure of the edge e. Let {e1, e2, e˜1, e˜2} ⊂ E \{e} be the set of edges such that

x1 ≺ ei ≺ fi, x2 ≺ e˜i ≺ fi, i = 1, 2.

Then any parallel neighbore ˜ of e satisﬁese ˜ 6∈ {e, e1, e2, e˜1, e˜2} and

x1 ≺ e,˜ or x2 ≺ e,˜ ore ˜ ≺ f1, ore ˜ ≺ f2. This yields that

#{parallel neighbors of e} = |x1| + |x2| + |f1| + |f2| − 12. Hence by (2), we get

RicF (e) = 2 + 2 − (|x1| + |x2| + |f1| + |f2| − 12). This proves the proposition. For any vertex x of degree N, we denote by

(|σ1|, |σ2|, ··· , |σN |) N the pattern of x, where {σi}i=1 are the faces incident to x, ordered by |σ1| ≤ |σ2| ≤ · · · ≤ |σN |. In the next proposition, we prove that for a graph G on a surface with positive Forman curvature, the medial graph G0 has positive combinatorial curvature everywhere.

Proposition 2.6. Let G be a graph on surface, and e be an edge with RicF (e) > 0. 0 Then the list of vertex patterns for Me in the medial graph G is given by (5) (3, 3, 3, 3), (3, 3, 3, 4), (3, 3, 3, 5), (3, 3, 3, 6), (3, 3, 4, 4), (3, 3, 4, 5), (3, 4, 4, 4). 0 In particular, Φ(Me) > 0. Hence, for any G ∈ FC+,G has positive combinatorial curvature. 8 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG

Proof. Let x1, x2 ∈ V and f1, f2 ∈ F satisfy x1 ≺ e, x2 ≺ e, e ≺ f1, e ≺ f2. Then by (4), |x1| + |x2| + |f1| + |f2| ≤ 15. 0 By the structure of the medial graph G , the facial degrees of faces incident to Me are given by |x1|, |x2|, |f1| and |f2|. See Figure 4. Since |xi| ≥ 3, |fi| ≥ 3, i = 1, 2, 0 we obtain the list of possible vertex patterns of Me in G as in (5). In particular, the case-by-case calculation yields Φ(Me) > 0. This proves the proposition.

e x2 x1

Figure 4. Medial graph G0 (dash) of a graph G (solid).

2 S For each p = 4, 5, 6, a p-gonal antiprism (Figure 5) is in FC+ .

Figure 5. p-gonal antiprism (p = 4, 5, 6).

We discuss the relation between graphs on S2 and on RP 2. For any graph G on RP 2, since π : S2 → RP 2 is a double cover, we can lift the graph structure G to S2, denoted by G.e Theorem 2.7. For any tessellation G on RP 2, the lifted graph Ge on S2 is a tessellation.

Proof. We verify the tessellation properties, see the introduction, for G.e The properties (i), (ii), (iii) are trivial. It is suﬃcient to prove (iv). That is, we need to prove the following: for any faces σ1, σ2 in G,e (a) if σ1 ∩ σ2 contains more than two points, then it is the closure of an edge, and (b) if σ1 ∩ σ2 is one point, then it is a vertex.

To prove (a), suppose it is not true, then there are two faces σ1, σ2 in Ge such that σ1 ∩ σ2 contains at least two vertices, but it is not the closure of an edge. Let 2 2 π : S → RP be the universal cover, and τi = π(σi), i = 1, 2. By the construction of the universal cover, we recall the following property: for any simply-connected GRAPHS ON SURFACES WITH POSITIVE FRRMAN OR CORNER CUVATURES 9

2 −1 open subset U in RP , π (U) consists of two disjoint subsets U1 and U2 such that

π|Ui : Ui → U (i = 1, 2) are homeomorphisms, see e.g. the proof in [19, pp. 64–65]. For any face τ in G, since τ is a closed disk, there exists an open neighborhood −1 A of τ such that π (A) = A1 t A2 and π|Ai (i = 1, 2) are homeomorphisms. We claim that τ1 6= τ2. Suppose it is not true, i.e. τ1 = τ2. Applying the above property for τ = τ1, we obtain homeomorphisms π|Ai : Ai → A, where A is an open neighborhood of τ1. Since σ1 ∩ σ2 6= ∅,

σ1 ∪ σ2 ⊂ A1 (or A2).

Without loss of generality, suppose that σ1 ∪ σ2 ⊂ A1, then (σ1 ∪ σ2) ∩ A2 = ∅. −1 This yields a contradiction since π (τ1) = σ1 ∪ σ2, and π|A2 is a homeomorphism. This proves the claim. For distinct points {p1, p2} ⊂ σ1 ∩ σ2, by the same argument as above for τ1, we get a homeomorphism between a neighborhood of σ1 and a neighborhood of τ1. This implies that π(p1) 6= π(p2). Hence τ1 ∩ τ2 consists of at least two points. By the tessellation properties of G, τ1 ∩ τ2 is the closure of an edge in G. Moreover, this yields that τ1 ∪ τ2 is homeomorphic to a closed disk. Then there exists an open −1 neighborhood W of τ1 ∪ τ2 such that π (W ) = W1 t W2 and π|Wi (i = 1, 2) are homeomorphisms. Since σ1 ∩ σ2 6= ∅,

σ1 ∪ σ2 ⊂ W1 (or W2).

Since π|W1 is a homeomorphism, by the property of τ1 ∩ τ2, σ1 ∩ σ2 is the closure of an edge in G.e This yields a contradiction and proves (a). To prove (b), for any faces σ1 and σ2 satisfying σ1 ∩ σ2 = {p}, by the same argument as above, one can show that τ1 6= τ2, where τi = π(σi), i = 1, 2. Hence τ1 ∩ τ2 6= ∅. We claim that τ1 ∩ τ2 is one point. Suppose that it contains more than two points, then it is the closure of an edge. Then the same argument as above yields that there exists a homeomorphism between an open neighborhood of τ1 ∪ τ2 and an open neighborhood of σ1 ∪ σ2. This contradicts σ1 ∩ σ2 = {p}. This proves the claim. Hence by the tessellation properties of G, τ1∩τ2 is a vertex. Then τ1∪τ2 is simply- connected and there exists a simply-connected, open neighborhood W of τ1 ∪ τ2. By the same argument as above, W is homeomorphic to an open neighborhood of σ1 ∪ σ2. This implies p is a vertex, and yields the result (b). This proves the theorem. 2 RP By the above theorem, in order to classify the class FC+ , it is suﬃcient to 2 S 2 classify the class FC+ and to ﬁgure out the candidates whose projections into RP are tessellations.

2.1. Embedding. To fully represent the embedded graph, we need both the abstract graph (V,E) and the cyclic edge orders. In the case of a graph with no multiedges, it is conventional to give both at once by listing neighbors in anti- clockwise order. An adjacency list of a graph G is, by deﬁnition, a list of pairs of a vertex x and the counter-clockwise cyclic list Nx of vertices adjacent to x. A mirror image of a graph G is, by deﬁnition, a graph such that an adjacency list is a list of pairs of a vertex x and reversed Nx. 10 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG

b a a b c c d d f e e f

a:bcde, b:caf, c:dab, d:eacf, e:dfa, f:bed a:edcb, b:fac, c:bad, d:fcae, e:afd, f:deb

Figure 6. A graph and an adjacency list. The left part and the right part are “mirror image” to each other.

Let G be the set of connected, simple, planar graphs G = (V,E,F ) such that both of face degrees and vertex degrees are at least 3 and finite. Given G, G0 ∈ G. G is said to be order-preserving isomorphic (OP-isomorphic, for short) to G0, if there is a bijection ϕ from G to G0 that preserves incidence relation for vertices, edges, and faces, the cyclic edge-orderings around the vertices. G is said to be order-reversing isomorphic (OR-isomorphic, for short) to G0, if G is OP-isomorphic to a mirror image of G0. G is said to be isomorphic to G0, if they are OP-isomorphic or OR-isomorphic [7]. “plantri is a program that generates certain types of graphs that are imbedded on the sphere. Exactly one member of each isomorphism class is output, using an amount of memory almost independent of the number of graphs produced. This, together with the exceptionally fast operation and careful validation, makes the program suitable for processing very large numbers of graphs. Exactly one member of each isomorphism class is output, using an amount of memory almost independent of the number of graphs produced. This, together with the exceptionally fast operation and careful validation, makes the program suitable for processing very large numbers of graphs. Isomorphisms are defined with respect to the imbeddings, so in some cases outputs may be isomorphic as abstract graphs.” (plantri- guide.txt [7]) Below, the figures of graphs G = (V,E,F ) are based on some embeddings ϕ’s to the plane such that • the edges of G become mutually noncrossing line segments of the figures; and • the cyclic ordering of edges of G incident to each vertex v ∈ V becomes the counter-clockwise ordering of the line segments around the point ϕ(v). For example, in Figure 7, the left is 3d-representation of a graph as a convex polytope, and the right is an embedding of the same graph such that the edges become mutually noncrossing line segments and the cyclic edge-ordering around each vertex is preserved. Jn stands for the n-th Johnson solid [24, 39].

3. Graphs on surfaces with positive Forman curvature

In this section, we give a new proof of Theorem 1.2 and classify the class FC+. GRAPHS ON SURFACES WITH POSITIVE FRRMAN OR CORNER CUVATURES 11

Figure 7

Proof of Theorem 1.2. Let G0 be the medial graph of G. Then G0 is a tessellation of S, see Proposition 2.3. Moreover, G0 has positive combinatorial curvature by Proposition 2.6. By DeVos-Mohar’s solution of Higuchi’s conjecture, Theorem 1.1, we prove the theorem.

In the following, we prove Theorem 1.3 by classifying the class FC+. We need the following lemmas.

0 0 0 0 Lemma 3.1. For any medial graph G = (V ,E ,F ) of G ∈ FC+, the vertex pattern of any v is given in (5), and #V 0 ≤ 24. Proof. The list of vertex patterns follows from Proposition 2.6. Moreover, for any vertex v, whose vertex pattern is given in the list, Φ(v) ≥ 1/12. By the discrete Gauss-Bonnet theorem (1), 1 X #V 0 ≤ Φ(v) ≤ 2. 12 v∈V 0 This yields the result. A graph G = (V,E,F ) on S2 is called a spherical quadrangulation if |f| = 4 for any f ∈ F. For any f ∈ F, the face pattern of f is given by (|v1|, |v2|, |v3|, |v4|), where 4 {vi}i=1 are vertices incident to f and |v1| ≤ |v2| ≤ |v3| ≤ |v4|. Let Q be the set of 2-connected, simple, spherical quadrangulations G = (V,E,F ) with #F ≤ 24, whose face patterns are in the list (5).

2 S 0 ∗ Lemma 3.2. For any G ∈ FC+ , (G ) ∈ Q. Proof. Since G0 is 4-regular, the lemma follows from Proposition 2.1, Proposi- tion 2.2, Proposition 2.3, and Lemma 3.1. 2 S Proof of Theorem 1.3. We first classify the finite set FC+ . By Lemma 3.2, for any 2 S 0 ∗ G ∈ FC+ , (G ) ∈ Q. We enumerate the class Q, which serves the set of candidates 2 0 ∗ S of (G ) for some G ∈ FC+ , by a computer program. Then using the duality, we obtain G0 from the output, and by Theorem 2.4, we construct the dual pair (G, G∗), whose medial graph is G0, in the canonical way (described after Theorem 2.4). This 2 S gives the classification of FC+ . The program is a modification of Brinkmann-McKay’s plantri. Plantri is a program that generates certain types of graphs that are embedded on the sphere. Exactly one member of each isomorphism class is output, using an amount of memory almost independent of the number of graphs produced. Here isomorphisms are graph isomorphisms which also take embedding to the plane (sphere) into account. In particular, plantri enumerates quickly all 4-regular H, such as medial graphs, by an algorithm developed in [6]. Every spherical quadrangulation of vertices more 12 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG

than 4 is obtained from a pseudo-double wheel through ﬁnite applications of two local expansions (see Figure 8).

Figure 8. p-gonal pseudo-double wheels (p = 3, 4, 5), and two expansions of spherical quadrangulations to increase the number of faces (Brinkmann et al. [6]).

We added to plantri, the following: (1) the restriction of vertex pattern of H to be one of (5); (2) the computation of the dual pair (G, G∗) such that H is the medial graph of (G, G∗). 2 0 S As a result, we found that the set of medial graphs G of FC+ -graphs consists of 73 simple, 2-connected, 4-regular, planar graphs. For each graph G0, we generate the dual pair (G, G∗) such that the G0 is the medial graph of (G, G∗). We check that all obtained pairs (G, G∗) are tessellations on S2. There are exactly 30 self- 2 ∗ ∗ S dual pairs. We say the pair (G, G ) is self-dual if G = G . Hence FC+ consists of 30 + (73 − 30) × 2 = 116 graphs. 2 2 2 S S S Among the 116 FC+ -graphs, only five graphs FC+ -E10-1V6, FC+ -E12-8V7, 2 2 2 S S S FC+ -E12-8V7Dual, FC+ -E13-1V7, and FC+ -E13-1V8Dual are not 3-connected. Hence, the five graphs are not skeletons of convex polyhedra, by Steinitz’s theorem [41]. 2 S The 116 FC+ -graphs are presented in Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, and Table 10. Graphs G = (V,E,F ) in each row of each table are ordered by #V . The rows in each table are ordered by #E,#V , the vertex-connectedness, and the self-duality of the first graph G = (V,E,F ) of the respective row. The second column of each table is for numbering rows among the same #E. For self-dual G, we do not present the dual G∗ in the table. Rows GRAPHS ON SURFACES WITH POSITIVE FRRMAN OR CORNER CUVATURES 13

#E # #V G = (V,E,F ) #F G∗ 6 1 4 Regular tetrahedron 8 1 5 Square pyramid (J1) 9 1 5 3-gonal bipyramid (J12) 6 3-gonal prism

10† 1 6

10 2 6 5-gonal pyramid (J2)

10 3 6

11 1 6 7

11 2 6 7

2 S Table 1. 11 FC+ -graphs such that 6 ≤ #E ≤ 11.

2 2 2 S S S FC+ -E10-1, FC+ -E12-8, and FC+ -E13-1 are indicated with †. In the three rows, each graph is not 3-connected. But in the other rows, each graph is 3-connected. 2 S In the tables of FC+ -graphs, a graph of i edges, No. j, k vertices is referred by 2 2 S S FC+ -Ei-jVk, and the dual graph by FC+ -Ei-jVkDual. 2 S By the classiﬁcation of FC+ , using Theorem 2.7, we can determine the class 2 2 RP S FC+ as follows: for any G ∈ FC+ we check whether there exists a ﬁxed-point free, involutive isomorphism of G and the according projection in RP 2 is a tessellation. 2 RP We show that FC+ = ∅. This completes the proof of Theorem 1.3.

4. Graphs on surfaces with positive corner curvature In this section, we prove Theorem 1.4.

Lemma 4.1. For any G = (V,E,F ) ∈ C+, we have the following: (1) Both of the vertex degrees and the facial degrees are 3,4,or 5, and the vertex pattern of any d-valent vertex is (3d) for any d = 4, 5. ∗ (2) G ∈ C+. By Lemma 4.1 (1), the minimum combinatorial curvature of the vertex pattern is 1/10. So the number of vertices is at most 20. By Lemma 4.1 (2), the number of faces is so. It also implies that if we get all the graphs in C+ whose #V ≤ #F , after dual operation we can get all the graphs in C+.

Theorem 4.2. For any G = (V,E,F ) ∈ C+, #V ≤ #F implies #V ≤ 12. Proof. By the Euler formula, #V ≤ #E/2 + 1. Because G is a tessellation, G has a medial graph G0 = (V 0,E0,F 0) and #E = #V 0, by Proposition 2.3. By Lemma 4.1 (1), the vertex pattern of G0 is (3, 3, k, l)(k, l ∈ {3, 4, 5}). The minimum 14 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG

#E # #V G = (V,E,F ) #F G∗ 12 1 6 Regular octahedron 8 Cube 12 2 7 Hexagonal pyramid

12 3 7

12 4 7

12 5 7 (Elongated triangular pyramid (J7))

12 6 7

12 7 7

12† 8 7 7

12 9 7 7

2 S Table 2. 12 12-edge FC+ -graphs.

combinatorial curvature of G0 is 1/15 given by (3, 3, 5, 5). Then #V 0 ≤ 2/(1/15) = 30. Hence, #V ≤ #E/2 + 1 = #V 0/2 + 1 = 16. Then we need to prove when 13 ≤ #V ≤ 16, G is not in C+. The number of vertices of degree d, the number of faces of facial degree d, the number of vertices, the number of edges, the number of faces, and the number of vertices of pattern p are denoted by vd, fd, v, e, f, and np respectively. By simple computation of a brute-force computer program, the only combinations of nonnegative integers v3, f3 (3 ≤ d ≤ 5), v, e, f, np (p = 333, 334, 335, 344, 345, 355, 444, 445, 455, 555) such that 5 5 5 X X X v = vd, f = fd, v − e + f = 2, 2e = dvd, d=3 d=3 d=3 v ≤ f, 13 ≤ v ≤ 16,

v3 = n333 + n334 + n335 + n344 + n345 + n355 + n444 + n445 + n455 + n555,

3f3 = 3n333 + 2n334 + 2n335 + n344 + n345 + n355 + 4v4 + 5v5,

4f4 = n334 + 2n344 + n345 + 3n444 + 2n445 + n455,

5f5 = n335 + n345 + 2n355 + n445 + 2n455 + 3n555, are the following 15 combinations: GRAPHS ON SURFACES WITH POSITIVE FRRMAN OR CORNER CUVATURES 15

#E # #V G = (V,E,F ) #F G∗

13† 1 7 8

13 2 7 8

13 3 7 8

13 4 7 8

13 5 7 8

13 6 7 8

13 7 7 8

2 S Table 3. 14 13-edge FC+ -graphs.

(1) v3=5,v4=1,v5=7,f3=13,f4=0,f5=3,v=13,e=27,f=16,n333=0,n334=0, n335=0,n344=0,n345=0,n355=0,n444=0,n445=0,n455=0,n555=5

(2) v3=7,v4=1,v5=5,f3=10,f4=0,f5=4,v=13,e=25,f=14,n333=0,n334=0, n335=0,n344=0,n345=0,n355=1,n444=0,n445=0,n455=0,n555=6

(3) v3=8,v4=1,v5=4,f3=8,f4=1,f5=4,v=13,e=24,f=13,n333=0,n334=0, n335=0,n344=0,n345=0,n355=0,n444=0,n445=0,n455=4,n555=4

(4) v3=8,v4=1,v5=4,f3=8,f4=1,f5=4,v=13,e=24,f=13,n333=0,n334=0, n335=0,n344=0,n345=0,n355=0,n444=0,n445=1,n455=2,n555=5

(5) v3=8,v4=1,v5=4,f3=8,f4=1,f5=4,v=13,e=24,f=13,n333=0,n334=0, n335=0,n344=0,n345=0,n355=0,n444=0,n445=2,n455=0,n555=6

(6) v3=8,v4=1,v5=4,f3=8,f4=1,f5=4,v=13,e=24,f=13,n333=0,n334=0, n335=0,n344=0,n345=0,n355=0,n444=1,n445=0,n455=1,n555=6

(7) v3=5,v4=0,v5=9,f3=15,f4=0,f5=3,v=14,e=30,f=18,n333=0,n334=0, n335=0,n344=0,n345=0,n355=0,n444=0,n445=0,n455=0,n555=5 16 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG

#E # #V G = (V,E,F ) #F G∗

14 1 7 9

14 2 7 9

14 3 8

14 4 8

14 5 8

14 6 8

14 7 8 8

(Gyrobifastigium (J26))

14 8 8 8

14 9 8 8

14 10 8 8

2 S Table 4. 16 14-edge FC+ -graphs.

(8) v3=7,v4=0,v5=7,f3=12,f4=0,f5=4,v=14,e=28,f=16,n333=0,n334=0, n335=0,n344=0,n345=0,n355=1,n444=0,n445=0,n455=0,n555=6

(9) v3=8,v4=0,v5=6,f3=10,f4=1,f5=4,v=14,e=27,f=15,n333=0,n334=0, n335=0,n344=0,n345=0,n355=0,n444=0,n445=0,n455=4,n555=4

(10) v3=8,v4=0,v5=6,f3=10,f4=1,f5=4,v=14,e=27,f=15,n333=0, n334=0,n335=0,n344=0,n345=0,n355=0,n444=0,n445=1,n455=2,n555=5

(11) v3=8,v4=0,v5=6,f3=10,f4=1,f5=4,v=14,e=27,f=15,n333=0, n334=0,n335=0,n344=0,n345=0,n355=0,n444=0,n445=2,n455=0,n555=6

(12) v3=8,v4=0,v5=6,f3=10,f4=1,f5=4,v=14,e=27,f=15,n333=0, GRAPHS ON SURFACES WITH POSITIVE FRRMAN OR CORNER CUVATURES 17

#E # #V G = (V,E,F ) #F G∗

15 1 7 10

(5-gonal bipyramid (J13)) (5-gonal prism)

15 2 8 9

15 3 8 9

15 4 8 9

15 5 8 9

15 6 8 9

15 7 8 9

15 8 8 9

2 S Table 5. 16 15-edge FC+ -graphs.

n334=0,n335=0,n344=0,n345=0,n355=0,n444=1,n445=0,n455=1,n555=6

(13) v3=9,v4=0,v5=5,f3=9,f4=0,f5=5,v=14,e=26,f=14,n333=0,n334=0, n335=0,n344=0,n345=0,n355=2,n444=0,n445=0,n455=0,n555=7

(14) v3=9,v4=0,v5=5,f3=9,f4=0,f5=5,v=14,e=26,f=14,n333=0,n334=0, n335=1,n344=0,n345=0,n355=0,n444=0,n445=0,n455=0,n555=8

(15) v3=10,v4=0,v5=6,f3=10,f4=0,f5=6,v=16,e=30,f=16,n333=0, n334=0,n335=0,n344=0,n345=0,n355=0,n444=0,n445=0,n455=0,n555=10

The last combination is rejected by the following Fact 1 (7), and the other combinations by Fact 1 (6). 18 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG

#E # #V G = (V,E,F ) #F G∗

16 1 8 10

16 2 8 10 (square antiprism) (4-gonal pseudo-double wheel)

16 3 9 (Elongated square pyramid (J8))

16 4 9

16 5 9

16 6 9

16 7 9

16 8 9

16 9 9 9

2 S Table 6. 12 16-edge FC+ -graphs.

Fact 1.

(6) v5 = 0, f5 = 0, or n335 > 0.

(7) n335 6= 1, n345 > 0, or n355 > 0.

Proof. (6) Assume v5 > 0 and f5 > 0. Then, there are a 5-valent vertex x and a 5-gon f. By Lemma 4.1 (1), for d = 4, 5, any d-valent vertex has type (3d). So, x is not a vertex of a 5-gon. Because of the connectedness, there is a shortest path P from x = x0, x1, x2, . . . , xn such that xn is a vertex of the 5-gon f. The edge 5 {x0, x1} is shared by two 3-gons, because x0 has type (3 ) by Lemma 4.1 (1). If n = 1, then the degree d1 of x1 is 3, by Lemma 4.1 (1). Hence n335 > 0. If n > 1, GRAPHS ON SURFACES WITH POSITIVE FRRMAN OR CORNER CUVATURES 19

#E # #V G = (V,E,F ) #F G∗

17 1 9 10

17 2 9 10

17 3 9 10

17 4 9 10

2 S Table 7. 8 17-edge FC+ -graphs.

then the degree d1 of x1 is 4 or 5, because P is a shortest path. By Lemma 4.1 (1), d the vertex pattern of x1 is (3 ). Hence, the edge {x1, x2} is shared by two 3-gons. By repeating this argument, each {xi−1, xi} (1 ≤ i ≤ n) is shared by two 3-gons. By Lemma 4.1 (1), xn has vertex type (3, 3, 5). Hence n335 > 0. (7) Assume n335 = 1 and n345 = 0. By n335 = 1, there is a unique vertex u of type (3, 3, 5). The two vertices adjacent to u in the 5-gon should have vertex patterns (3, x, 5) and (3, y, 5) for some x 6= 3 and some y 6= 3. This establishes Fact 1.

Therefore, for any G = (V,E,F ) ∈ C+, if #V ≤ #F , then #V ≤ 12. This completes the proof of Theorem 4.2. Proof of Theorem 1.4. By Lemma 4.1 and Theorem 4.2, we enumerate the finite set C+ by a computer program. We first classify planar graphs in C+. The program is a modification of Brinkmann-McKay’s plantri. Their program plantri can enumerate all the simple, 2-connected, planar graphs such that all facial degree is greater than 2 and less than 6 and the number v of vertices is a given number. We modify plantri so that any vertex pattern is one of (3d)(d = 3, 4, 5), (3, 3, 4), (3, 3, 5), (3, 4, 4), (3, 4, 5), (3, 5, 5), (4, 4, 4), (4, 4, 5), (4, 5, 5), (5, 5, 5), and v is at most the number of faces, based on Lemma 4.1 (1),(2). Then, we run the modified plantri with the number of vertices 4,..., 12. It outputs 13 graphs. By Lemma 4.1 (2), we take the dual graphs into account. Then we obtain 22 simple, 2-connected, planar graphs such that corner curvature is positive everywhere. Table 11 and Table 12 present the 22 graphs. There are one 4-edge self-dual graph, one 8-edge self-dual graph, and one 10-edge self-dual graph. The other 19 graphs are not self-dual. All the 22 planar graphs of C+ are actually 3-connected. From the classification of planar graphs in C+, see Table 11 and Table 12, by 2 Theorem 2.7, we obtain the classification of graphs on RP in C+. By similar case- by-case checking as in the proof of Theorem 1.3, it consists of hemi-icosahedron 20 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG

#E # #V G = (V,E,F ) #F G∗

18 1 9 11 (medial graph of triangular prism)

18 2 9 11

18 3 10

18 4 10

18 5 10

18 6 10

18 7 10 10

18 8 10 10

2 S Table 8. 12 18-edge FC+ -graphs.

and hemi-dodecahedron, see e.g. [29], which are the projections of the regular icosahedron and the regular dodecahedron on S2 into RP 2 via the double cover 2 2 π : S → RP .

References [1] Akama, Y., Hua, B., Su, Y., Wang, L.: A curvature notion for planar graphs stable under planar duality. arXiv:1909.07825 (2019) [2] Archdeacon, D.: The medial graph and voltage-current duality. Discrete Math. 104, 111–141 (1992) [3] Baues, O., Peyerimhoﬀ, N.: Curvature and geometry of tessellating plane graphs. Discrete Comput. Geom. 25(1), 141–159 (2001). GRAPHS ON SURFACES WITH POSITIVE FRRMAN OR CORNER CUVATURES 21

#E # #V G = (V,E,F ) #F G∗

19 1 10 11

19 2 10 11

19 3 10 11

20 1 11

20 2 11

(Elongated 5-gonal pyramid (J9))

20 3 11

2 2 S S Table 9. 6 19-edge FC+ -graphs, and 3 20-edge FC+ -graphs.

Figure 9. The only simple, 2-connected, projective planar graphs with positive corner curvature: Hemi-dodecahedron (left) and hemi-icosahedron (right). In each, arrows with the same style are identiﬁed.

[4] Baues, O., Peyerimhoﬀ, N.: Geodesics in non-positively curved plane tessellations. Adv. Geom. 6(2), 243–263 (2006). [5] Biggs, N.: Algebraic graph theory, second edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1993) [6] Brinkmann, G., Greenberg, S., Greenhill, C., McKay, B.D., Thomas, R., Wollan, P.: Gener- ation of simple quadrangulations of the sphere. Discrete Math. 305(1-3), 33–54 (2005). 22 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG

#E # #V G = (V,E,F ) #F G∗

21 1 11 12

22 1 12

24 1 13

24 2 12 14

(Cuboctahedron) (Rhombic dodecahedron) 2 2 S S Table 10. 2 21-edge FC+ -graphs, 1 22-edge FC+ -graphs, and 3 2 S 24-edge FC+ -graphs.

[7] Brinkmann, G., McKay, B.: Guide to using plantri (version 5.0) (2016). URL http://cs. anu.edu.au/~bdm/plantri. Accessed on 17 January 2020 [8] Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry, Graduate Studies in Math- ematics, vol. 33. American Mathematical Society, Providence, RI (2001). [9] Burago, Y., Gromov, M., Perelman, G.: A. D. Aleksandrov spaces with curvatures bounded below. Russian Math. Surveys 47(2), 1–58 (1992) [10] Chen, B.: The Gauss-Bonnet formula of polytopal manifolds and the characterization of embedded graphs with nonnegative curvature. Proc. Amer. Math. Soc. 137(5), 1601–1611 (2009). [11] Chen, B., Chen, G.: Gauss-Bonnet formula, finiteness condition, and characterizations of graphs embedded in surfaces. Graphs Combin. 24(3), 159–183 (2008). [12] Conway, J.H., Burgiel, H., Goodman-Strauss, C.: The symmetries of things. A K Peters, Ltd., Wellesley, MA (2008) [13] DeVos, M., Mohar, B.: An analogue of the Descartes-Euler formula for infinite graphs and Higuchi’s conjecture. Trans. Amer. Math. Soc. 359(7), 3287–3300 (2007). [14] Deza, M., Dutour Sikirić,M.: Geometry of chemical graphs: polycycles and two-faced maps, Encyclopedia of Mathematics and its Applications, vol. 119. Cambridge University Press, Cambridge (2008). [15] Forman, R.: Bochner’s method for cell complexes and combinatorial Ricci curvature. Discrete Comput. Geom. 29(3), 323–374 (2003). [16] Ghidelli, L.: On the largest planar graphs with everywhere positive combinatorial curvature. arXiv:1708.08502 (2017) [17] Gromov, M.: Hyperbolic groups. In: Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, pp. 75–263. Springer, New York (1987). [18] Häggström, O., Jonasson, J., Lyons, R.: Explicit isoperimetric constants and phase transi- tions in the random-cluster model. Ann. Probab. 30(1), 443–473 (2002). [19] Hatcher, A.: Algebraic topology. Cambridge University Press, Cambridge (2002) GRAPHS ON SURFACES WITH POSITIVE FRRMAN OR CORNER CUVATURES 23

#E # #V G = (V,E,F ) #F G∗ 4 1 4 Regular tetrahedron 4 8 1 5 Square pyramid (J1) 5 3-gonal 9 1 5 5 3-gonal prism bipyramid (J12) 5-gonal 10 1 6 6 pyramid (J2)

12 1 6 8

(Biaugmented tetrahedron) 12 1 6 Regular octahedron 8 Cube 5-gonal 15 1 7 10 5-gonal prism bipyramid (J13) Table 11. All the 22 2-connected, simple, planar graphs such that every corner curvature is positive (1/2). The graphs in the 2 S ﬁrst four, the sixth, and the seventh lines are in FC+ , but the two graphs in the ﬁfth line are not.

[20] Higuchi, Y.: Combinatorial curvature for planar graphs. J. Graph Theory 38(4), 220–229 (2001). [21] Higuchi, Y., Shirai, T.: Isoperimetric constants of (d, f)-regular planar graphs. Interdiscip. Inform. Sci. 9(2), 221–228 (2003). [22] Hua, B., Jost, J., Liu, S.: Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature. J. Reine Angew. Math. 700, 1–36 (2015). [23] Ishida, M.: Pseudo-curvature of a graph. In: Lecture at Workshop on topological graph theory. Yokohama National University (1990) [24] Johnson, N.W.: Convex polyhedra with regular faces. Canad. J. Math. 18, 169–200 (1966). [25] Keller, M.: The essential spectrum of the Laplacian on rapidly branching tessellations. Math. Ann. 346(1), 51–66 (2010). [26] Keller, M.: Curvature, geometry and spectral properties of planar graphs. Discrete Comput. Geom. 46(3), 500–525 (2011). [27] Keller, M., Peyerimhoff, N.: Cheeger constants, growth and spectrum of locally tessellating planar graphs. Math. Z. 268(3-4), 871–886 (2011). [28] Lawrencenko, S., Plummer, M.D., Zha, X.: Isoperimetric constants of infinite plane graphs. Discrete Comput. Geom. 28(3), 313–330 (2002). [29] McMullen, P., Schulte, E.: Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, vol. 92. Cambridge University Press, Cambridge (2002). [30] Mohar, B., Thomassen, C.: Graphs on surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD (2001) [31] Nevanlinna, R.: Analytic functions. Translated from the second German edition by Phillip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162. Springer-Verlag, New York-Berlin (1970) [32] Oh, B.G.: On the number of vertices of positively curved planar graphs. Discrete Math. 340(6), 1300–1310 (2017). [33] Oldridge, P.R.: Characterizing the polyhedral graphs with positive combinatorial curvature. Master’s thesis, Department of Computer Science, University of Victoria (2017). URL http: //hdl.handle.net/1828/8030. Accessed on 17 January 2020 24 YOHJI AKAMA, BOBO HUA, YANHUI SU, AND HAOHANG ZHANG

#E # #V G = (V,E,F ) #F G∗

15 2 7 10

(Augmented octahedron)

18 1 8 12 (Snub 3-gonal prism) (Gyroelongated [14, p.20] triangular bipyramid)

18 2 8 12

(Snub disphenoid (J84))

21 1 9 13

(Triaugmented 3-gonal prism (J51))

24 1 10 16 (Snub square prism) (Gyroelongated square [14, p.20] bipyramid (J17)) Regular 30 1 12 Regular icosahedron 20 dodecahedron Table 12. All the 22 2-connected, simple, planar graphs such that every corner curvature is positive (2/2). All the graphs in this table 2 S are not in FC+ .

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[38] Woess, W.: A note on tilings and strong isoperimetric inequality. Math. Proc. Camb. Phil. Soc. 124(3), 385–393 (1998) [39] Zalgaller, V.A.: Convex polyhedra with regular faces. Translated from Russian. Seminars in Mathematics, V. A. Steklov Mathematical Institute, Leningrad, Vol. 2. Consultants Bureau, New York (1969) [40] Zhang, L.: A result on combinatorial curvature for embedded graphs on a surface. Discrete Math. 308(24), 6588–6595 (2008). [41] Ziegler, G.M.: Lectures on polytopes, Graduate Texts in Mathematics, vol. 152. Springer- Verlag, New York (1995) [42] Zuk,˙ A.: On the norms of the random walks on planar graphs. Ann. Inst. Fourier (Grenoble) 47(5), 1463–1490 (1997).

(Yohji Akama) Mathematical Institute, Graduate School of Science, Tohoku Univer- sity, Sendai, 980-0845, Japan., Tel.: +81-22-795-6402, Fax: +81-22-795-6400 E-mail address: [email protected]

(Bobo Hua) School of Mathematical Sciences, LMNS, Fudan University, Shanghai 200433, China; Shanghai Center for Mathematical Sciences, 2005 Songhu Road, Shanghai 200438, China E-mail address: [email protected]

(Yanhui Su) College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China E-mail address: [email protected]

(Haohang Zhang) Shanghai Center for Mathematical Sciences, Fudan University, Shang- hai 200433, China E-mail address: [email protected]